The Hilbert-Riesz Wavelet Transform and the Multiresolution Analysis of Fringe Patterns

M. Unser

In optics, the 2D generalization of the Hilbert transform is known as the radial Hilbert transform or the spiral phase quadrature transform. It is closely related to the Riesz transform which has the remarkable property of mapping a real 2D wavelet basis into a complex one. We propose to use such a Riesz pair of polyharmonic spline wavelet transforms to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We also propose a corresponding wavelet-domain procedure for estimating the underlying instantaneous frequency of the signal. We illustrate the method with the analysis of 2D fringe patterns.